lucataco/numinamath-7b-tir 🔢📝 → 📝
About
NuminaMath is a series of language models that are trained to solve math problems using tool-integrated reasoning (TIR)
Example Output
"For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?"
Output
Understand the Polynomial and its Roots:
The polynomial given is (x^2 + kx + 36).
Let the roots of the polynomial be (r_1) and (r_2).Relationships Between Roots and Coefficients:
For a polynomial (x^2 + bx + c = 0), the sum of the roots (r_1 + r_2 = -b) and the product of the roots (r_1 \cdot r_2 = c).
In our case, (r_1 + r_2 = -k) and (r_1 \cdot r_2 = 36).Conditions for Distinct Integer Roots:
We need to find pairs of distinct integers ((r_1, r_2)) such that their product is 36. Then, for each pair, we will determine the corresponding value of (k).Check for Distinct Pairs:
Ensure (r_1
eq r_2).Calculate the Values of (k):
For each valid pair ((r_1, r_2)), compute (k = -(r_1 + r_2)) and count the distinct values.
Let's implement this in Python to find the number of distinct values of (k).
import sympy as sp
# Define the product of the roots
product = 36
# Find all pairs of integers (r1, r2) such that r1 * r2 = product
integer_pairs = []
for i in range(1, product + 1):
if product % i == 0:
j = product // i
integer_pairs.append((i, j))
integer_pairs.append((j, i)) # Include both (i, j) and (j, i)
# Filter pairs to ensure they are distinct and r1 != r2
distinct_pairs = [(r1, r2) for r1, r2 in integer_pairs if r1 != r2]
# Calculate the corresponding values of k
k_values = set(-(r1 + r2) for r1, r2 in distinct_pairs)
# Output the number of
Performance Metrics
All Input Parameters
{
"top_k": 50,
"top_p": 0.9,
"prompt": "For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?",
"max_tokens": 512,
"min_tokens": 0,
"temperature": 0.6,
"system_prompt": "You are a helpful assistant.",
"presence_penalty": 0,
"frequency_penalty": 0
}
Input Parameters
- top_k
- The number of highest probability tokens to consider for generating the output. If > 0, only keep the top k tokens with highest probability (top-k filtering).
- top_p
- A probability threshold for generating the output. If < 1.0, only keep the top tokens with cumulative probability >= top_p (nucleus filtering). Nucleus filtering is described in Holtzman et al. (http://arxiv.org/abs/1904.09751).
- prompt
- Prompt
- max_tokens
- The maximum number of tokens the model should generate as output.
- min_tokens
- The minimum number of tokens the model should generate as output.
- temperature
- The value used to modulate the next token probabilities.
- system_prompt
- System prompt to send to the model. This is prepended to the prompt and helps guide system behavior. Ignored for non-chat models.
- stop_sequences
- A comma-separated list of sequences to stop generation at. For example, '<end>,<stop>' will stop generation at the first instance of 'end' or '<stop>'.
- presence_penalty
- Presence penalty
- frequency_penalty
- Frequency penalty
Output Schema
Output
Example Execution Logs
INFO 07-12 13:42:28 async_llm_engine.py:646] Received request ed4c5fab43324c8b8e97700a86a10dea: prompt: '### Problem: For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?\n### Solution: ', params: SamplingParams(n=1, best_of=1, presence_penalty=0.0, frequency_penalty=0.0, repetition_penalty=1.0, temperature=0.6, top_p=0.9, top_k=50, min_p=0.0, seed=None, use_beam_search=False, length_penalty=1.0, early_stopping=False, stop=[], stop_token_ids=[100001], include_stop_str_in_output=False, ignore_eos=False, max_tokens=512, min_tokens=0, logprobs=None, prompt_logprobs=None, skip_special_tokens=True, spaces_between_special_tokens=True, truncate_prompt_tokens=None), prompt_token_ids: None, lora_request: None.
stdoutINFO 07-12 13:42:32 metrics.py:295] Avg prompt throughput: 7.6 tokens/s, Avg generation throughput: 26.5 tokens/s, Running: 1 reqs, Swapped: 0 reqs, Pending: 0 reqs, GPU KV cache usage: 0.3%, CPU KV cache usage: 0.0%.
stdoutINFO 07-12 13:42:37 metrics.py:295] Avg prompt throughput: 0.0 tokens/s, Avg generation throughput: 43.4 tokens/s, Running: 1 reqs, Swapped: 0 reqs, Pending: 0 reqs, GPU KV cache usage: 0.6%, CPU KV cache usage: 0.0%.
stdoutGeneration took 1720790260.82sFormatted prompt: ### Problem: For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?
### Solution: INFO 07-12 13:42:40 async_llm_engine.py:134] Finished request ed4c5fab43324c8b8e97700a86a10dea.
stdout
Version Details
- Version ID
3749451b2f11e1cd14cc3adc1b66a17356c0f3529780382f3245f486908bd215- Version Created
- July 12, 2024